When carrying out non-linear processing of sampled data with well-defined, limited bandwidths, the harmonics produced by components close to the Nyquist sampling limit result in aliasing, that is, frequencies that fold back into the passband of the system. Because these folded-back frequencies cannot be removed by post-filtering, they cause artifacts that degrade the final processed data. A good example of non-linear processing is in television scanning systems, where the transfer characteristic between the driving signal into a cathode ray tube and the light given out by the tube is non-linear. Indeed, the light output by the tube is closely proportional to the input raised to a constant power. This power-law constant is called the gamma of the display tube, and the process of correction, basically to multiply the signal by the inverse of the gamma constant, is called gamma correction.
In an analog television system, the analog gamma correction circuit ordinarily has adequate bandwidth to handle the harmonic frequencies generated. In a digital television system, gamma correction is desirably implemented digitally, typically by using the input video to address a read-only memory (used as a look-up memory) that has been programmed with a table of output values based upon the non-linear gamma function. Each input video sample functions as an address for producing a corrected output video sample according to the function. Due to the limited bandwidth of a digital system and to its sampled nature, however, any harmonics produced above the Nyquist limit will be folded back to produce artifacts in the final picture. The normal Nyquist requirement is to sample the analog input signal at a frequency that is at least twice the highest possible input frequency. This will prevent aliasing at the input. However, a subsequent non-linear operation, such as gamma correction, performed on the sampled signals produces harmonic components, especially upper harmonics, that fold back into the signal spectrum, even though the initial sampling of the input analog signal met the Nyquist requirement.
Current methods to overcome this problem include de-emphasizing the high frequency components prior to the non-linear process and then to re-emphasize these components after the process. An example of this method is described in International Publication No. WO 91/03122, which was published Mar. 7, 1991 in the name of Rank Cintel, Limited. The problem with this method is that the harmonics are only reduced, and not eliminated, Another well-known approach to this problem up-converts the sampled data by interpolation to, e.g., double the effective samples and thereby to increase the effective bandwidth of the system while keeping the bandwidth of the data the same. While this method produces accurate results, the problem is with the much increased complexity in the signal processing.